Original connector, generally speaking,referred to big name connectors, such as Molex, TYCO, JST, JAE, Deutsch, etc. As these companies has been in this field for many years, famous for its high quality and continuous creativity.
Equivalent connector, also called replacement connector, compatible connector, cross connector, in most cases, referred to the connectors produced in China. As these manufacturers do not deal with connector filed for that long, not as famous as Molex, TYCO, JST, Deutsch.
Original connector and equivalent connector share the same specifications, including temperature, pitch, material, appearance, etc.
And of course they have their difference, such as cost, delivery, quality, etc. Original connector is more expensive than equivalent connector and delivery is longer. Quality of original connector is always considered the best. And not all original connector has corresponding equivalent.